Computing using unknown values

ABSTRACT

A method of computing includes defining a first atomic random variable (ARV) and first random variable (RV) in a programming language system. The first ARV having a non-deterministic value of either zero according to a second probability or one according to a first probability. A sum of the first probability and the second probability is one. A covariance of the first ARV and a second ARV is zero. The first RV has a first indefinite value at a first definite probability and includes a polynomial of one or more atomic random variables (ARVS) that includes the first ARV. The method includes executing a computer instruction that includes a mathematical operation involving the first RV as a basic data type and produces a second RV having a second indefinite value at a second definite probability, represents a result distribution, and tracks a response to the one or more ARVS.

FIELD

The embodiments discussed herein are related to computing using unknownvalues.

BACKGROUND

Computer programmers translate real-world problems into steps ofcomputer programs or computer instructions. The computer programs areprocessed by computer devices, which are generally arithmetic machines.The computer programs are written in computer programming languages.

Computer programming languages recognize different types of data. Thetype(s) of data recognized by a computer language may be referred to asa basic data type. In JAVA for instance, the basic (or primitive) datatypes include byte, short, int, long, float, double, Boolean, and char.In C, the basic data types may include int, float, double, char. Theseand other basic data types represent a single state at a time.

Values and variables in computer programs are formatted in the basicdata type. Accordingly, values and variables in the computer programsrepresent a single state that is known. The computer programs mayaccordingly only perform computer operations using known values. Forexample, in the computer programs, operations such as an additionoperation and multiplication operation may be executed on known values.Similarly, variables in the conventional programming languages may haveknown values during execution of computer operations. The basic datatype being single state at a time and being known introduces alimitation of the computer programs and the computer languages.

The subject matter claimed herein is not limited to embodiments thatsolve any disadvantages or that operate only in environments such asthose described above. Rather, this background is only provided toillustrate one example technology area where some embodiments describedherein may be practiced.

SUMMARY

According to an aspect of an embodiment, a method of quantum-inspiredcomputing may include defining a first atomic random variable (ARV) in aprogramming language system. The first ARV may have a non-deterministicvalue of either zero or one. The first ARV may have a first probabilityof having a value of one. The first ARV may have a second probability ofhaving a value of zero. A sum of the first probability and the secondprobability may be equal to one. A covariance of the first ARV and asecond ARV may be zero. The method may include defining a first randomvariable (RV) in the programming language system. The first RV may havea first indefinite value at a first definite probability and may includea polynomial of one or more atomic random variables (ARVS) that includesthe first ARV. The method may include executing a computer instructionthat includes a mathematical operation. The mathematical operation mayinvolve the first RV as a basic data type. The execution of the computerinstruction may produce a second RV that has a second indefinite valueat a second definite probability and the second RV that may represent aresult distribution and tracks response to the one or more ARVS.

The object and advantages of the embodiments will be realized andachieved at least by the elements, features, and combinationsparticularly pointed out in the claims.

It is to be understood that both the foregoing general description andthe following detailed description are exemplary and explanatory and arenot restrictive of the invention, as claimed.

BRIEF DESCRIPTION OF THE DRAWINGS

Example embodiments will be described and explained with additionalspecificity and detail through the use of the accompanying drawings inwhich:

FIG. 1 illustrates an example programming language system in which someembodiments described in the present disclosure may be implemented;

FIG. 2 illustrates an example computing system configured forquantum-inspired computing in the programming language system of FIG. 1;

FIG. 3 is a flow diagram of an example method of quantum-inspiredcomputing;

FIG. 4 presents an example equation that defines a ground of an atomicrandom variable (ARV) defined and supported in the programming languagesystem of FIG. 1;

FIG. 5 presents an example equation that defines a mean of the ARV ofFIG. 4;

FIG. 6 presents an example equation that defines a mean square of theARV of FIG. 4;

FIG. 7 presents an example equation that defines a variance of the ARVof FIG. 4;

FIG. 8 presents example equations that define a covariance of the ARV ofFIG. 4;

FIG. 9 presents an example equation that defines an n-th moment of theARV of FIG. 4;

FIG. 10 presents an example equation that defines a probability densityfunction of the ARV of FIG. 4;

FIG. 11 presents an example equation that defines a cumulative densityfunction of the ARV of FIG. 4;

FIG. 12 presents an example equation that defines a moment-generatingfunction of the ARV of FIG. 4;

FIG. 13 presents an example equation that defines a characteristicfunction of the ARV of FIG. 4;

FIG. 14 presents an example equation that defines a cumulant-generatingfunction of the ARV of FIG. 4;

FIG. 15 presents a theorem of power of the ARV of FIG. 4;

FIG. 16 presents an example equation that defines a ground of a linearrandom variable (LRV) defined and supported in the programming languagesystem of FIG. 1;

FIG. 17 presents an example equation that defines a mean of the LRV ofFIG. 16;

FIG. 18 presents an example equation that defines a mean square of theLRV of FIG. 16;

FIG. 19 presents an example equation that defines a variance of the LRVof FIG. 16;

FIG. 20 presents an example equation that defines n-th moment of the LRVof FIG. 16;

FIG. 21 presents example equations that define a covariance of the LRVof FIG. 16;

FIG. 22 presents an example equation that defines a correlationcoefficient of the LRV of FIG. 16;

FIG. 23 presents example quadratic random variable (QRV) transformexpressions that transform a QRV defined and supported in theprogramming language system of FIG. 1;

FIG. 24 presents an example equation that defines a ground of the QRV ofFIG. 23;

FIG. 25 presents an example equation that defines a mean of the QRV ofFIG. 23;

FIG. 26 presents an example equation that defines a mean square of theQRV of FIG. 23;

FIG. 27 presents an example equation that defines a variance of the QRVof FIG. 23;

FIGS. 28A, 28B, and 29 presents example equations that define acovariance of the QRV of FIG. 23;

FIG. 30 presents an example equation that defines a correlationcoefficient of the QRV of FIG. 23;

FIGS. 31-32 presents equations for an example transformation between theQRV of FIG. 23 and the LRV of FIG. 16;

FIG. 33 presents some example changes to the ground after thetransformation of FIGS. 31-32;

FIG. 34 presents some example changes to the mean after thetransformation of FIGS. 31-32;

FIGS. 35A and 35B present some example changes to the mean square afterthe transformation of FIGS. 31-32;

FIG. 36A presents an example high-order random variable (HRV)transformation expressions that transform an HRV defined and supportedin the programming language system of FIG. 1 to an equivalent canonicalform;

FIGS. 36B and 36C present a proof of the HRV transformation expressionsof FIG. 36A;

FIG. 37 presents an example equation that defines a ground of an HRVdefined and supported in the programming language system of FIG. 1;

FIG. 38 presents an example equation that defines a mean of the HRV ofFIG. 37;

FIG. 39 presents an example equation that defines a mean square of theHRV of FIG. 37;

FIG. 40 presents an example equation that defines a variance of the HRVof FIG. 37;

FIGS. 41A-41C present example equations that define a covariance of theHRV of FIG. 37;

FIG. 42 presents an example equation that defines a correlationcoefficient of the HRV of FIG. 37;

FIGS. 43A-43B present an example reduction of the HRV of FIG. 37 byassignment of local value;

FIGS. 44A-44C present some example changes to the equations of theground after the reduction of FIGS. 43A-43B;

FIGS. 45A and 45B present some example changes to the equations of themean after the reduction of FIGS. 43A-43B;

FIGS. 45C-45D present equations clarifying portions of the equations ofFIGS. 45A and 45B;

FIGS. 46A-46E present some example changes to the equations of the meansquare (e.g., the second moment) after the reduction of FIGS. 43A-43B;

FIG. 47 presents some example differences to the mean square due to thereduction of FIGS. 43A-43B;

FIG. 48 presents an example random variable (RV) subtraction operationdefined and supported in the programming language system of FIG. 1;

FIG. 49 presents an example RV scaling operation defined and supportedin the programming language system of FIG. 1;

FIG. 50 presents example RV multiplication operations that involve LRVSdefined and supported in the programming language system of FIG. 1;

FIG. 51 presents example RV multiplication operations that involve QRVSdefined and supported in the programming language system of FIG. 1;

FIG. 52 presents an example definition for conditional random variables(CRVS) for the LRVS defined and supported in the programming languagesystem of FIG. 1;

FIG. 53 presents an example definition for the CRVS for the QRVS definedand supported in the programming language system of FIG. 1;

FIG. 54 presents an example definition for the CRVS for the HRVS definedand supported in the programming language system of FIG. 1;

FIGS. 55A and 55B present an example definition for RV derivativeoperations for the LRVS defined and supported in the programminglanguage system of FIG. 1;

FIGS. 56A and 56B present an example definition for RV derivativeoperations for the QRVS;

FIGS. 57A and 57B present an example definition for RV derivativeoperations for the HRVS;

FIGS. 58A-58C present example expressions that defines a ground of theRV derivatives;

FIG. 59 presents an example expression that defines an LRV mean of theRV derivatives;

FIG. 60 presents example expressions that define a QRV mean of the RVderivatives;

FIG. 61 presents example expressions that define an HRV mean of the RVderivatives;

FIG. 62 presents an example expression that defines an LRV variance ofthe RV derivatives;

FIGS. 63A and 63B present example expressions that define a QRV varianceof the RV derivatives;

FIGS. 64A-64D present example expressions that define an HRV variance ofthe RV derivatives;

FIG. 65 presents an example expression that defines an LRV covariance ofthe RV derivatives;

FIGS. 66A-66C present example expressions that define a QRV covarianceof the RV derivatives;

FIG. 67 presents example expressions for entropy of the ARV;

FIGS. 68A-68C present example expressions for entropy of the LRV;

FIGS. 69A-69C present example expressions for entropy of the QRV; and

FIGS. 70A-70C present example expressions for entropy of the HRV,

arranged in accordance with at least one embodiment described herein.

DESCRIPTION OF EMBODIMENTS

Current basic data types of computer programming languages may representa single state at a time. Accordingly, values and variables in computerprograms that implement the current basic data types represent a singlestate at a time. In these and other computer programming languages, thesingle state is known during execution of mathematic operations includedin computer programs.

The basic data type being a single state at a time and being known is alimitation of the computer programs and of the computer languages.Indeed, many real-word problems include unknown values. For example,quantum simulation for drug design, quantum simulation for materialdesign, and combinatorial optimization for place and route incomputer-aided design (CAD) may involve equations using many unknownvalues. To implement a computer program that addresses a real-worldproblem that includes unknown values, programmers may use algebraicmathematics with unknown variables to solve the equations beforehand, oruse a numerical algorithm to translate equation that describe thereal-world problems with the unknown values to simple arithmeticoperations that include only known values.

The numerical algorithm is often inefficient. In particular, whendimensions of the real-world problem is high, numerical algorithm may beinefficient because the numerical algorithm may involve checking a largenumber (e.g., billions of billions) of combinations of possible values.Furthermore, the number of combinations may grow exponentially as thenumber of unknown values increases.

In addition, some statistical analysis involve random variables. Forexample, statistical analysis in statistical eye simulation of signalintegrity analysis for high-speed signal transmission or statisticalstatic timing analysis for CAD involve random variables. Conventionalprogramming languages (e.g., those that only support basic data typeswith known values and/or a single state at a time) do not support randomvariables because the random variables may have unknown values.Accordingly, it may be difficult and error prone to write computerprograms for statistical analysis in conventional programming languages.Indeed, some equations involved in statistical analysis may beimpossible to solve. Additionally, a ‘bug’ in such a computer program isoften indistinguishable from a result variation. For instance, jitterand inter-symbol interference in signal-integrity analysis may becorrelated with each other because jitter and inter-symbol interferenceare both data dependent. However, a correlation between jitter andinter-symbol interference is difficult to describe, hardly noticeable,and overlooked in equations.

Some other computing systems may be able to represent multiple states ata time. For example, quantum computers, in some implementations, may beable to represent multiple states at a time. The quantum computer mayutilize quantum effects of quantum devices. However, the quantumcomputer is difficult to realize itself. Furthermore, it may bedifficult to write an algorithm and a computer program for the quantumcomputer because the programming model is different from conventionalvon Neumann-style computer devices and computer programming languages.

Another example is a non-deterministic algorithm, which may be able torepresent a single state randomly chosen from multiple states at a time.For instance, a Monte Carlo (MC) method and a Markov-Chain Monte Carlo(MCMC) method may be able to represent a single state randomly chosenfrom multiple states at a time. The MC method explores solution space ina non-deterministic way using pseudo random numbers instead of exploringthe entire solution space to relax computing inefficiencies associatedwith increases in dimensionality. However, in the non-deterministicalgorithm results are obtained only by chance. Additionally, thenon-deterministic algorithm is inefficient to cover low-probabilitycases especially in circumstance of high dimensionality.

Symbolic computation systems may support unknown values or randomvariables. Symbolic computation systems are incorporated in softwarepackages such as REDUCE™, MATHEMATICA®, or MAPLE™. Computation insymbolic computation is performed using symbols for unknown values inthe same way as conventional algebra used by human being. However,symbolic computation is not always possible. For example, it ismathematically proved impossible to solve a fifth-order equation insymbolic computation. In circumstances in which the symbolic computationis not possible, the computation may switch to numerical computation. Asdiscussed above, the numerical computation is arithmetic. After thesymbolic computation is transitioned to the numerical computation,computation for unknown values discontinues. For instance, correlationbetween random variables is lost after a transition to numericalcomputation. There is no intermediate form of computation betweensymbolic computation and numeric computation.

Statistical analysis software may support random variables. For example,in statistical analysis software such as ‘S’ or ‘R,’ random variablesare used to represent a set of statistical data. However, thestatistical analysis software is generally implemented for statisticalanalysis on the given data set. Accordingly, the statistical analysissoftware does not support general-purpose programming.

Satisfiability (SAT) solvers may be implemented to check whether a logicexpression, which is represented with logic variables with conjunction;disjunction; and invert operators, is satisfiable. The SAT solvers mayfind the assignment of each logic variable when it is satisfiable. Thelogic variables are regarded to have unknown logic values. Thus, the SATsolver may be regarded to perform logic computation on the unknown logicvalues. There are heuristics for the SAT solvers that enable thesoftware to solve logic expressions transformed from combinatorialoptimization problems. However, SAT solvers may solve only logicexpressions. Thus, problems are transformed to a logic expression beforeapplication of the SAT solver. Accordingly, the SAT solver isrestrictive compared to the general-purpose computer programming.

Accordingly, embodiments described in the present disclosure relate togeneral-purpose computing using unknown values. For example, someembodiments may relate to computing systems that use unknown values as abasic data type, but are implemented in a general-purpose computingsystem and enable the generation and programming of general purposecomputer programs.

An example of general purpose computing using unknown values may includequantum-inspired computing. The quantum-inspired computing may bedefined relative to a quantum computer. A quantum computer uses aquantum bit (qubit) that represents either 0 or 1. In quantum computerswhether the qubit is 0 or 1 is unknown. Multiple qubits compose aquantum state that represents one of super-positioned values, but whichof the super-positioned values is unknown. Multiple qubits may be alsoentangled in a quantum state. Entanglement between the multiple qubitsin the quantum state is realizable only if qubits are implemented usingquantum device which has quantum effects, and may not be not realizableusing conventional device which does not has quantum effects.

Some embodiments described in the present disclosure are related toquantum-inspired computing, because it realizes the super-position ofmultiple possible values without realizing the entanglement in a quantumstate. Accordingly, it is possible to implement using conventionaldevice which does not have quantum effects or as a software on aconventional general-purpose computer system.

Some embodiments describe a programming language system. The programminglanguage system may support a random variable as a basic data type. Therandom variable may have an indefinite value, which may include adefinite probability. The random variable may be represented by anatomic random variable that may have an indefinite atomic value that mayhave a definite probability distribution. The indefinite atomic valuemay have a discrete atomic value that may be either zero or one or mayhave one value from 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 or may have acontinuous atomic value. The continuous atomic value may have a Gaussianprobability distribution or a uniform probability distribution.

The random variable may be represented by a polynomial of the atomicrandom variable. The programming language system may support multiplemathematical operations and statistical properties that enablenavigation of the solution space. For an example, a value may be chosento assign to an atomic random variable, which may equal to 0 or 1, basedon whether an expected value of a derivative of a random variable withrespect to the atomic random variable is positive or negative. Inanother example, a list of atomic random variables may be sorted in thedescending order of variance of a derivative of a random variable.Accordingly, the list from those atomic random variables may beprocessed with larger effects on the random variable to those atomicrandom variables with smaller effects on the random variable.

Some example mathematical operations and statistical properties mayinclude a product operator on the random variable, a conditionalmodifier on the random variable, a derivative modifier on the randomvariable as well as other, similar mathematical operations.Additionally, the statistical properties may include mean, mean square(2^(nd) moment), variance, n-th moment, covariance, correlationcoefficient, ground, other statistical properties, or some combinationthereof. The programming language system may be loaded with anapplication program and may be executed.

The programming language system may be implemented to write and executeprograms for various applications. For example, the applications mayinclude quantum simulation for drug design, quantum simulation formaterial design, combinatory optimization for place and route in CAD,statistical eye simulation of signal integrity analysis for high-speedsignal transmission, and statistical static timing analysis for CAD.

In comparison to statistical analysis software, the programming languagesystem may be suitable for general-purpose programming. In comparison toSAT solvers, because the programming language system is based onpolynomial, it is more general, more flexible, and more suitable forgeneral-purpose programming. For instance, a polynomial may representany logic formula by representing True by 1, False by 0, an invertingoperator by a subtract operator (1−X), a conjunction operator by aproduct operator (X*Y), and a disjunction operator by a few operators(X+Y−X*Y).

These and other embodiments are described with reference to the appendedfigures. In the appended figures, features and components with the sameitem number include similar structure and function unless otherwisespecified.

FIG. 1 illustrates an example programming language system 100 in whichsome embodiments described in the present disclosure may be implemented.In the programming language system 100, a client device 102 and/or aserver device 106 may be configured for implementation of a computerprogram module 108. The computer program module 108 may enable a user112 to program or otherwise generate computer instructions 105, whichmay take the form of a computer program. The computer program module 108may include a general-purpose programming language that may beimplemented to generate the computer instructions 105.

The computer instructions 105 may include generally applicableprogramming instructions that may be implemented to provide any type ofcomputing functionality. For instance, the computer instructions 105 mayinclude von Neumann-style computer instructions. The computerinstructions 105 may be executed in an application module 107 togenerate results 113. The results 113 may include a computing output, adata output, a signal that includes a data output, etc.

In the programming language system 100 of FIG. 1, the computer programmodule 108 and/or the application module 107 may be implemented at theclient device 102. For example, the computer program module 108 may beloaded at least partially on the client device 102. Additionally, theapplication module 107 may be loaded at least partially on the clientdevice 102. The user 112 may generate the computer instructions 105locally (e.g., on the client device 102) and/or execute the computerinstructions 105 locally.

Additionally, one or both of the computer program module 108 and theapplication module 107 may be implemented on the server device 106. Forinstance, the computer program module 108 and/or the application module107 may be accessed by the user 112 using the client device 102 via anetwork 104. In some embodiments, the computer instructions 105 may becommunicated between the client device 102, the server device 106, orother devices that are capable of network communication via the network104. Each of the client device 102, the server device 106, and thenetwork 104 are described below.

The network 104 may be wired or wireless, and may have numerousdifferent configurations including, but not limited to, a starconfiguration, token ring configuration, or other configurations.Furthermore, the network 104 may include a local area network (LAN), awide area network (WAN) (e.g., the Internet), and/or otherinterconnected data paths across which multiple devices may communicate.In some embodiments, the network 104 may be a peer-to-peer network. Thenetwork 104 may also be coupled to or include portions of atelecommunications network that may enable communication of data in avariety of different communication protocols. In some embodiments, thenetwork 104 includes BLUETOOTH® communication networks and/or cellularcommunications networks for sending and receiving data including viashort messaging service (SMS), multimedia messaging service (MMS),hypertext transfer protocol (HTTP), direct data connection, wirelessapplication protocol (WAP), e-mail, etc.

The client device 102 may be a computing device or system that includesa processor, memory, and network communication capabilities. Forexample, the client device 102 may include a laptop computer, a desktopcomputer, a smart phone, a tablet computer, a mobile telephone, apersonal digital assistant (“PDA”), a mobile email device, or otherelectronic device capable of accessing the network 104.

The server device 106 may include a hardware server that includes aprocessor, memory, and communication capabilities. In the illustratedembodiments, the server device 106 may be coupled to the network 104 tosend and receive data and information to and from the client device 102via the network 104.

The computer program module 108 and/or the application module 107 may beimplemented using hardware including a processor, a microprocessor(e.g., to perform or control performance of one or more operations), afield-programmable gate array (FPGA), or an application-specificintegrated circuit (ASIC). In some other instances, the computer programmodule 108 and/or the application module 107 may be implemented using acombination of hardware and software. Implementation in software mayinclude rapid activation and deactivation of one or more transistors ortransistor elements such as may be included in hardware of a computingsystem (e.g., the client device 102 and/or the server device 106).Additionally, software defined instructions may operate on informationwithin transistor elements. Implementation of software instructions mayat least temporarily reconfigure electronic pathways and transformcomputing hardware.

In the programming language system 100, the user 112 may be associatedwith the client device 102. The user 112 may interface with the computerprogram module 108 and/or the application module 107 using the clientdevice 102 or a subsystem thereof as well as a combination of the clientdevice 102, the network 104, and the server device 106. The user 112 mayinclude an individual. In some embodiments, the user 112 may includeanother entity, a group of users, a robot, or a computing system.

The computer instructions 105 that are generated by the computer programmodule 108 may include one or more mathematical operations. Themathematical operations in the computer instructions 105 may involve arandom variable (in the singular, “RV” and in the plural “RVS”) as abasic data type. Accordingly, the computer instructions 105 mayrecognize the RV as a data type that is executable by the applicationmodule 107. The RV may be defined in the computer program module 108.The RV may include a polynomial of one or more atomic random variables(in the singular, “ARV” and in the plural “ARVS”).

The ARVS may be defined in the computer program module 108. The RV mayhave an indefinite value or a non-deterministic value at a definiteprobability. For example, in some embodiments, the ARV may have anon-deterministic value of either zero or one. In these and otherembodiments, the ARV may have a first probability of having a value ofone and a second probability of having a value of zero. A sum of thefirst probability and the second probability is one. In someembodiments, the ARV may have a non-deterministic value that may be acontinuous atomic value. The continuous atomic value may have a Gaussianprobability distribution or a uniform probability distribution, forinstance. A covariance between the ARVS is zero.

The computer program module 108 may support and define multiple types ofRVS. For example, the computer program module 108 may support and definea linear random variable (in the singular, “LRV” and in the plural“LRVS”), a quadratic random variable (in the singular, “QRV” and in theplural “QRVS”), and high-order random variables (in the singular, “HRV”and in the plural “HRVS”). In addition, conditional random variables (inthe singular, “CRV” and in the plural “CRVS”) may be defined andsupported by the computer program module 108. Some additional details ofLRVS, QRVS, HRVS, and CRVS are provided elsewhere in the presentdisclosure.

The execution of mathematical operations that involve the RVS mayproduce new RVS that best represents a result distribution, keepingtrack of response to the original ARVS. The mathematical operations thatmay be included in the computer instructions 105 and are supported anddefined by the computer program module 108 may include an RV additionoperation, an RV subtraction operation, an RV scaling operation, an RVmultiplication operation, and an RV derivative operation. Someadditional details of each of the RV addition operation, the RVsubtraction operation, the RV scaling operation, the RV multiplicationoperation, and the RV derivative operation are provided elsewhere in thepresent disclosure.

In addition, the computer program module 108 may define statisticalproperties of the mathematical operations, ARVS, and RVS. Thestatistical properties may enable navigation of a solution space thatresults or may result from execution of the mathematical operations, forinstance.

Accordingly, the computer program module 108 may be configured to definethe RVS as a basic data type, the ARVS, the mathematical operators, andthe statistical properties in the programming language system 100. Thecomputer program module 108 may be configured to enable the computerinstructions 105 to be written using the RVS, the ARVS, the mathematicaloperators, and the statistical properties. The computer instructions 105may be executed by the application module 107.

The following paragraphs provide some additional details of each of theARVS, the LRVS, the QRVS, the HRVS, linear operators, the CRVS, and thederivative operators.

ARVS

The computer program module 108 may define ARVS. The ARVS may providethe basis for the RVS. For example, the RVS in the computer instructions105 may include a polynomial of one or more of the ARVS. Some examplesof the polynomial are provided elsewhere in the present disclosure.

In some embodiments, the ARVS may take a non-deterministic value ofeither one or zero. In computer science, “non-deterministic” refers to apotentially different output or value with the same input. Accordingly,with relation to the ARVS, the value of the ARVS is not exactly knownfor at least a period of time. The ARVS may take the non-deterministicvalue according to one or more probabilities. For instance, aprobability of the ARVS eventually having the value of one may bedefined as a first probability. A probability of the ARVS eventuallyhaving the value of zero may be defined as a second probability. A sumof the first probability and the second probability may be equal to one.Accordingly, the probability of the ARVS having the value of zero may beequal to a difference between one and the first probability. In thecomputer program module 108, the first probability and the secondprobability may be fixed at a particular time in which the ARV iscreated.

The ARVS may be independent of one another. Accordingly, instance of afirst ARV having a particular value (e.g., one or zero) may not affectvalues of a second ARV. In some embodiments, probabilities of ARVS maybe correlated. For instance, a second probability of a second ARV may bemanipulated based on a first probability of a first ARV.

In some embodiments, the computer program module 108 may define specialARVS. The special ARVS may include a first particular ARV that isassociated with a first probability that is equal to one. The firstparticular ARV may be equivalent to constant one and may be defined asone for all instances. The special ARVS may include a second particularARV that is associated with a first probability that is equal to zero.The second particular ARV may be equivalent to constant zero and may bedefined as zero for all instances.

In some embodiments, the ARV may be defined according to example ARVexpressions:X _(i)∈{0,1};0≤p _(i)≤1;Pr[X _(i)=1]=p _(i);Pr[X _(i)=0]=1−p _(i); andCov(X _(i) ,X _(k))=0 for i≠k.In the ARV expression, i and k represent indexing variables. Theparameter p_(i) represents the first probability indexed according tothe indexing variable i. The parameter X_(i) represents an ARV indexedaccording to the indexing variable i. X_(k) represents an ARV indexedaccording to the indexing variable k. The function Pr[ ] represents aprobability function. The function Cov( ) represents a covariancefunction.

According to the ARV expressions, the ARVS take a non-deterministicvalue of either one or zero. Determination of the value (either 1 or 0)may be deferred as much as possible until it becomes necessary. Forexample, it may become necessary when a number of the ARVS in a termexceeds a maximum order of the polynomial of random variables. At thispoint, values of one or more ARVS may be determined so that the numberof ARVS in a term does not exceed the maximum order of the polynomial ofRVs. Additionally, values of ARVS may have to be determined at a time toproduce the output of the program execution.

The determination may be deferred to enable a tracking of a correlation.The correlation may be between RVs that share the same ARV may betracked as long as the ARV is kept as an ARV without determining itsvalue. The correlation between the RVs that share the same ARV may belost after a value of the ARV is determined.

In some embodiment, the ARVS may have a continuous atomic value. Thecontinuous atomic value may have a Gaussian probability distribution ora uniform probability distribution.

The computer program module 108 may define multiple statisticalproperties. The statistical properties for the ARVS may include aground, a mean, a mean square, a variance, a covariance, an n-th moment,a probability density function, a cumulative density function, amoment-generating function, a characteristic function, and acumulant-generating function. In addition, the computer program module108 may define a theorem of power of ARV. The statistical properties mayenable navigation of a solution space.

In some embodiments, one or more of the mean, the mean square, and then-th moment of the ARVS may be equal to the first probability.Additionally, the mean, the mean square, and the n-th moment of the ARVSmay be equal to the first probability regardless of “n” of the n-thmoment. Additionally, the variance of the ARVS may be equal to theproduct of the first probability and the second probability(p_(i)(1−p_(i))). Additionally still, the m-th power of the ARVS may beequivalent to the ARVS themselves. The relationship between the m-thpower of the ARVS and the ARVS is because the n-th moment of the ARVSmay be equal to the first probability.

FIGS. 4-14 provide equations that may define statistical properties ofthe ARVS in some embodiments. In FIGS. 4-14, the parameters, functions,etc. are as defined above in the ARV expressions. The equations of FIGS.4-14 implement standard mathematical nomenclature and functionality,which may be understood by one with skill in the art with the benefit ofthe present disclosure.

FIG. 4 presents equation 1 (eq. 1). Equation 1 may define a ground(“G[parameter]” throughout the present disclosure) of the ARVS. Theground of the ARVS may be based on a value of a first probability. Forinstance in Equation 1, if the first probability is less than 0.5, thenthe ground may be equal to zero. Additionally, if the first probabilityis greater than 0.5, then the ground may be equal to one. Additionallystill, if the first probability may be equal to 0.5, the ground may beequal to 0.5.

FIG. 5 presents equation 2 (eq. 2). Equation 2 may define a mean(“E[parameter]” throughout the present disclosure) of the ARVS. FIG. 6presents equation 3 (eq. 3). Equation 3 may define a mean square(“E[parameter²]” throughout the present disclosure) of the ARVS. FIG. 7presents equation 4 (eq. 4). Equation 4 may define a variance(“V[parameter]” throughout the present disclosure) of the ARVS.

FIG. 8 presents equations 5-7 (eq. 5, eq. 6, and eq. 7). Equations 5-7may define a covariance (“Cov[parameter]” throughout the presentdisclosure) of the ARVS. In FIG. 8 (and other figures below), thevariable p₁ may represent a first probability of X₁ being equal to 1 andp₂ may represent a first probability of X₂ being equal 1. FIG. 9presents equation 8 (eq. 8). Equation 8 may define an n-th moment(“E[parameter^(n)]” throughout the present disclosure) of the ARVS.

FIG. 10 presents equation 9 (eq. 9). Equation 9 may define a probabilitydensity function of the ARVS. In FIG. 10, δ(x) is a Dirac's deltafunction. Some additional details of Dirac's delta function areavailable at https://en.wikipedia.org/wiki/Dirac_delta_function, whichis incorporated herein by reference. In FIG. 10 (and other figuresbelow), the variable x represents a parameter of a probability densityfunction ƒ_(X) _(i) (x) or other functions.

FIG. 11 presents equation 10 (eq. 10). Equation 10 may define acumulative density function of the ARVS. In FIG. 11, U(x) represents aunit step function. FIG. 12 presents equation 11 (eq. 11). Equation 11may define a moment-generating function M_(X) _(i) (t) of the ARVS. InFIG. 12 (and other figures below), the variable t represents a parameterof a moment-generating function M_(X) _(i) (t) or other functions. FIG.13 presents equation 12 (eq. 12). Equation 12 may define acharacteristic function φ_(X) _(i) (t) of the ARVS. FIG. 14 presentsequation 13 (eq. 13). Equation 13 may define a cumulant-generatingfunction K_(X) _(i) (t) of the ARVS. FIG. 15 presents a theorem of powerof the ARVS. The theorem illustrates that a power of an ARV isstatistically equivalent to the ARV itself. FIG. 15 presents equations14 and 15 (eq. 14 and eq. 15). Equation 14 defines the theorem statementand equation 15 provides the proof of equation 14.

LRVS

As discussed above, the RVS defined and supported in the programminglanguage system 100 may include the LRVS. The LRVS may include a linearsum of one or more ARVS. The LRVS may include a linear sum of aparticular number of the ARVS. The LRVS may further include adistribution of up to two raised to the power of the particular numberpossible values corresponding to zero or one of each of the ARVS.

In some embodiments, the LRVS may be defined. The LRV may be definedaccording to example LRV expressions:

${{Y_{a} \equiv {\sum\limits_{k = 0}^{m}\;{a_{k}X_{k}}}} = {{a_{0}X_{0}} + {a_{1}X_{1}} + {a_{2}X_{2}} + \cdots + {a_{m}X_{m}}}};$a_(k) ∈ ℝ  (or  ℂ); m ∈ ℕ; and p₀ = 1.In the LRV expressions, the parameter Y_(a) represents the LRV. Theparameter X_(k) represents an ARV indexed according to the indexingvariable k. The parameter a_(k) represents a deterministic real orcomplex value that represents a response to the ARV indexed according tothe indexing variable k. An example of the response to the ARV may bea_(k) of Y_(a) is the response of Y_(a) to X_(k), because a_(k) is thedifference between Y_(a) with X_(k) is equal to 1 and Y_(a) with X_(k)is equal to 0.

represents the set of real numbers.

represents the set of complex numbers.

represents the set of natural numbers. The value a₀X₀ is reserved forthe constant term with p₀=1 such that X₀ is equal to 1. The parameter p₀represents the first probability when k is equal to 0. The correlationwith other RVS is maintained through common ARVS.

The computer program module 108 may define multiple statisticalproperties for the LRVS. The statistical properties may enablenavigation of a solution space. The statistical properties for the LRVSmay include a ground, a mean, a mean square, a variance, an n-th moment,a covariance, and a correlation coefficient. FIGS. 16-22 provideequations that may define the statistical properties of the LRVS in someembodiments. In FIGS. 16-22, the parameters, functions, etc. are asdefined above in the LRVS expressions. The equations of FIGS. 16-22implement standard mathematical nomenclature and functionality, whichmay be understood by one with skill in the art with the benefit of thepresent disclosure.

FIG. 16 presents equation 16 (eq. 16). Equation 16 defines a ground(e.g. “G[Y_(a)]”) of the LRVS. The ground is a unique or substantiallyunique statistical attribute of the RVS. The ground may be close to themean value (described below). For instance, in circumstances in whichthe probabilities of ARVS are evenly distributed above 0.5 and below0.5, the ground may be close to the mean value. The ground may beaverage under a unique assumption to the values of ARVS. The ground maybe the value of the RV when the ARVS are either 0 or 1, which may dependon whether its probability is less than or more than 0.5 and the meanvalue is a value of the RV when all ARVS have their respective meanvalues. The ground may retrieve information from an entangled state.However, retrieval of the information from the entangled state may losecoherence. The ground may be useful in addition to the mean value andthe variance in quantum-inspired computing.

FIG. 17 presents equation 17 (eq. 17). Equation 17 may define a mean ofthe LRVS. FIG. 18 presents equation 18 (eq. 18). Equation 18 may definea mean square of the LRVS. FIG. 19 presents equation 19 (eq. 19).Equation 19 may define a variance of the LRVS. FIG. 20 presents equation20 (eq. 20). Equation 20 may define n-th moment of the LRVS. FIG. 21presents equations 21-23 (eq. 21, eq. 22, and eq. 23). Equations 21-23define a covariance of the LRVS. FIG. 22 presents equation 24 (eq. 24).Equation 24 may define a correlation coefficient of the LRVS.

QRVS

As discussed above, the RVS defined and supported in the programminglanguage system 100 may include the QRVS. The QRV may be definedaccording to example QRV expressions:

${Y_{A} \equiv {{\overset{\rightarrow}{X}}^{T}A\overset{\rightarrow}{X}}};$${\overset{\rightarrow}{X} \equiv \left\lbrack {X_{0}\mspace{14mu} X_{1}\mspace{14mu} X_{2}\mspace{11mu}\cdots\mspace{11mu} X_{m}} \right\rbrack^{T}};$${A \equiv \begin{bmatrix}a_{0,0} & a_{0,1} & a_{0,2} & \ldots & a_{0,m} \\a_{1,0} & a_{1,1} & a_{1,2} & \ldots & a_{1,m} \\a_{2,0} & a_{2,1} & a_{2,2} & \ldots & a_{2,m} \\\vdots & \vdots & \vdots & \ddots & \vdots \\a_{m,0} & a_{m,1} & a_{m,2} & \ldots & a_{m,m}\end{bmatrix}};$ a_(k, l) ∈ ℝ  (or  ℂ); m ∈ ℕ; and p₀ = 1.In the QRV expression, l represents an indexing variable. The parameterY_(A) represents the QRV. The parameter X_(k) represents an ARV indexedaccording to the indexing variable k. The parameter a_(k,l) represents adeterministic real or complex value that represents a response to theARV indexed according to the indexing variables k and l.

represents the set of real numbers.

represents the set of complex numbers.

represents the set of natural numbers. The parameter a₀X₀ is reservedfor the constant term with p₀=1 such that X₀ is equal to 1.

The QRV may be transformed to an equivalent canonical form. In someembodiments, the QRV may be transformed using QRV transform expressions,which are provided in FIG. 23.

The computer program module 108 may define multiple statisticalproperties for the QRVS. The statistical properties may enablenavigation of a solution space. The statistical properties for the QRVSmay include a ground, a mean, a mean square, a variance, a covariance,and a correlation coefficient. FIGS. 24-30 provide equations that mayapply to the QRVS in some embodiments for one or more of the statisticalproperties. In FIGS. 24-30, the parameters, functions, etc. are asdefined above in the QRVS expressions. The equations of FIGS. 24-30implement standard mathematical nomenclature and functionality, whichmay be understood by one with skill in the art with the benefit of thepresent disclosure.

FIG. 24 presents equation 25 (eq. 25). Equation 25 may define a groundof the QRVS. The ground of the QRVS is similar to the ground describedwith reference to LRVS. FIG. 25 presents equation 26 (eq. 26). Equation26 may define a mean of the QRVS. FIG. 26 presents equation 27 (eq. 27).Equation 27 may define a mean square of the QRVS. FIG. 27 presentsequation 28 (eq. 28). Equation 28 may define a variance of the QRVS.FIGS. 28A, 28B, and 29 presents equations 29-31 (eq. 29, eq. 30, and eq.31). Equations 29-31 define the covariance of the QRVS. FIG. 30 presentsequation 32 (eq. 32). Equation 32 define a correlation coefficient ofthe QRVS.

The LRV may be transformed to an equivalent QRV with diagonalcoefficient matrix. For example, FIG. 31 presents equations for anexample transformation from an LRV to a QRV.

The QRV may be transformed to an approximate LRV by partial assignmentof local values to ARVS. For example, FIG. 32 presents equations for anexample transformation from a QRV to an LRV. Additionally, FIGS. 33-35Bpresent changes to the equations above after the transformation. Forexample, FIG. 33 presents changes to the equations of the ground afterthe transformation. FIG. 34 presents changes to the equations of themean after the transformation. FIGS. 35A and 35B present changes to theequations of the mean square after the transformation.

HRVS

As discussed above, the RVS defined and supported by the in theprogramming language system 100 may include the HRVS. The HRV may bedefined according to example HRV expressions:

${Y_{A} \equiv {\sum\limits_{k_{1} = 0}^{m}\;{\sum\limits_{k_{2} = 0}^{m}\;{\ldots\mspace{11mu}{\sum\limits_{k_{d} = 0}^{m}\;{a_{k_{1},k_{2},\cdots,k_{d}}X_{k_{1}}X_{k_{2}}\mspace{11mu}\ldots\mspace{14mu} X_{k_{d}}}}}}}};$a_(k₁, k₂, ⋯ k_(d)) ∈ ℝ  (or  ℂ); m ∈ ℕ; d ≤ m; and p₀ = 1.In the HRV expressions, the parameter Y_(A) represents the HRV. Theparameter k_(i) represents an indexing variable. The parameter X_(k)_(i) represents an ARV indexed according to the indexing variable k_(i).The parameter a_(k) ₁ _(, k) ₂ _(, . . . , k) _(d) represents adeterministic real or complex value that represents a response to theARVS indexed according to the indexing variables k₁, k₂, . . . , k_(d).

represents the set of real numbers.

represents the set of complex numbers.

represents the set of natural numbers. The parametera_(0, 0, . . . , 0)X₀X₀ . . . X₀ is reserved for the constant term withp₀=1 such that X₀ is equal to 1. The parameter d represents the order ofthe HRV, that also represents the maximum number of ARVS in a singleterm. In the HRV expressions, if d is equal to 1, the HRV is a LRV andif d is equal to 2, the HRV is a QRV. The parameter m is a total numberof ARVS in a program excluding X₀ that is equal to 1 with theprobability of p₀=1. Accordingly, there are m number of ARVS from X₁through X_(m).

The HRV may be transformed to an equivalent canonical form. For example,the HRV may be transformed according to example HRV transformationexpressions. FIG. 36A presents the HRV transformation expressions. FIGS.36B and 36C present a proof of the HRV transformation expressions. InFIGS. 36A-36C, the parameters, functions, etc. are as defined above inthe HRV expressions. In the canonical form, an order of an HRV is lessthan or equal to m.

The computer program module 108 may define multiple statisticalproperties for the HRVS. The statistical properties for the HRVS mayinclude a ground, a mean, a mean square, a variance, a covariance, and acorrelation coefficient. FIGS. 37-42 provide equations that may apply tothe HRVS in some embodiments for one or more of the statisticalproperties. In FIGS. 37-42, the parameters, functions, etc. are asdefined above in the HRVS expressions. The equations of FIGS. 37-42implement standard mathematical nomenclature and functionality, whichmay be understood by one with skill in the art with the benefit of thepresent disclosure.

FIG. 37 presents equation 33 (eq. 33). Equation 33 may define a groundof the HRVS. The ground of the HRVS is similar to the ground describedwith reference to LRVS. FIG. 38 presents equation 34 (eq. 34). Equation34 may define a mean of the HRVS. FIG. 39 presents equation 35 (eq. 35).Equation 35 may define a mean square of the HRVS. FIG. 40 presentsequation 36 (eq. 36). Equation 36 may define a variance of the HRVS.FIGS. 41A-41C present equations 37-39 (eq. 37, eq. 38, and eq. 39).Equations 37-39 define a covariance of the HRVS. FIG. 42 presentsequation 40 (eq. 40). Equation 40 defines a correlation coefficient ofthe HRVS.

An order of HRVS may be reduced. For example, the order of HRVS may bereduced by assignment of local value. An example reduction of the HRVSis provided in FIGS. 43A-43B. In FIGS. 43A-43B, the parameters,functions, etc. are as defined above in the HRVS expressions. Theequations of FIGS. 43A-43B implement standard mathematical nomenclatureand functionality, which may be understood by one with skill in the artwith the benefit of the present disclosure.

Additionally, FIGS. 44A-48 present changes to the equations above afterthe reduction. For example, FIGS. 44A-44C present changes to theequations of the ground after the reduction. FIGS. 45A-45D presentchanges to the equations of the mean after the reduction. FIGS. 45A-45Bpresent the change to the equations. FIGS. 45C-45D present equationsclarifying portions of the equations of FIGS. 45A-45B.

FIGS. 46A-46E present changes to the equations of the mean square (e.g.,the second moment) after the reduction. FIGS. 46A-46B presents a checkof a case in which m=3, d=3, and r=2. FIG. 46C presents changes to theequations of the mean square after the reduction. FIGS. 46D and 46Epresents equations clarifying portions of the equations of FIG. 46C.FIG. 47 presents a difference to the mean square due to the reduction.

Linear Operators

The mathematical operations that may be included in the computerinstructions 105 and are supported and defined by the computer programmodule 108 may include an RV addition operation. The mathematicaloperation includes an RV addition operation of a first RV with a secondRV. Execution of the computer instruction produces a third RV that mayhave a particular indefinite value at a particular definite probabilityand the particular RV represents a result distribution and tracksresponse to the one or more ARVS.

In some embodiments, the RV addition operation is defined according toRV addition operation expressions:+:Y _(A) +Y _(B) →Y _(C); such that:

${Y_{A} = {a_{0} + {\sum\limits_{k_{1} = 1}^{m}\;{a_{k_{1}}X_{k_{1}}}} + {\sum\limits_{k_{1} = 1}^{m - 1}\;{\sum\limits_{k_{2} = {k_{1} + 1}}^{m}\;{a_{k_{1},k_{2}}X_{k_{1}}X_{k_{2}}}}} + \ldots + {\sum\limits_{k_{1} = 1}^{m - d_{A} + 1}\;{\sum\limits_{k_{2} = {k_{1} + 1}}^{m - d_{A} + 2}\;{\sum\limits_{k_{3} = {k_{2} + 1}}^{m - d_{A} + 3}\;{\ldots{\sum\limits_{k_{d_{A}} = {k_{d_{A - 1}} + 1}}^{m}\;{a_{k_{1},{k_{2}\cdots},k_{d_{A}}}X_{k_{1}}X_{k_{2}}\mspace{11mu}\ldots\mspace{11mu} X_{k_{d_{A}}}}}}}}}}};$${Y_{B} = {b_{0} + {\sum\limits_{k_{1} = 1}^{m}\;{b_{k_{1}}X_{k_{1}}}} + {\sum\limits_{k_{1} = 1}^{m - 1}\;{\sum\limits_{k_{2} = {k_{1} + 1}}^{m}\;{b_{k_{1},k_{2}}X_{k_{1}}X_{k_{2}}}}} + \ldots + {\sum\limits_{k_{1} = 1}^{m - d_{B} + 1}\;{\sum\limits_{k_{2} = {k_{1} + 1}}^{m - d_{B} + 2}\;{\sum\limits_{k_{3} = {k_{2} + 1}}^{m - d_{B} + 3}\;{\cdots{\sum\limits_{k_{d_{B}} = {k_{d_{B - 1}} + 1}}^{m}\;{b_{k_{1},{k_{2}\cdots},k_{d_{B}}}X_{k_{1}}X_{k_{2}}\mspace{11mu}{\ldots X}_{k_{d_{B}}}}}}}}}}};$${Y_{C} \equiv {c_{0} + {\sum\limits_{k_{1} = 1}^{m}\;{c_{k_{1}}X_{k_{1}}}} + {\sum\limits_{k_{1} = 1}^{m - 1}\;{\sum\limits_{k_{2} = {k_{1} + 1}}^{m}\;{c_{k_{1},k_{2}}X_{k_{1}}X_{k_{2}}}}} + \ldots + {\sum\limits_{k_{1} = 1}^{m - d_{C} + 1}\;{\sum\limits_{k_{2} = {k_{1} + 1}}^{m - d_{C} + 2}\;{\sum\limits_{k_{3} = {k_{2} + 1}}^{m - d_{C} + 3}\;{\cdots{\sum\limits_{k_{d_{C}} = {k_{d_{C - 1}} + 1}}^{m}\;{C_{k_{1},{k_{2}\cdots},k_{d_{C}}}X_{k_{1}}X_{k_{2}}\mspace{11mu}\ldots\mspace{11mu} X_{k_{d_{C}}}}}}}}}}};$     d_(A), d_(B) ≤ m;      d_(C) ≡ max (d_(A), d_(B));     a_(k₁, k₂, ⋯ k_(d_(A))) ∈ ℝ  (or  ℂ);     b_(k₁, k₂, ⋯ k_(d_(B))) ∈ ℝ  (or  ℂ);      m ∈ ℕ;     c_(o) ≡ a₀ + b₀; and$\mspace{76mu}{c_{k_{1},{k_{2}\cdots},k_{l}} \equiv \left\{ \begin{matrix}{a_{k_{1},{k_{2}\cdots},k_{l}} + b_{k_{1},{k_{2}\cdots},k_{l}}} & {{{for}\mspace{14mu}\left( {1 \leq l \leq d_{A}} \right)} ⩓ \left( {1 \leq l \leq d_{B}} \right)} \\a_{k_{1},{k_{2}\cdots},k_{l}} & {{for}\mspace{14mu}\left( {d_{B} < l \leq d_{A}} \right)} \\b_{k_{1},{k_{2}\cdots},k_{l}} & {{for}\mspace{14mu}\left( {d_{A} < l \leq d_{B}} \right)}\end{matrix} \right.}$In the RV addition operation, Y_(A), Y_(B), and Y_(C) represents theRVS. The Y_(A), Y_(B), and Y_(C) are high-order random variable (HRV).The parameters d_(A), d_(B), and d_(C) represent the order of Y_(A),Y_(B), and Y_(C), respectively. The parameters k₁ . . . k_(d) _(A)represent indexing variables. The parameter

X_(k) _(i) represents an ARV indexed according to the indexing variablek_(i). The parameters a_(k) ₁ _(, k) ₂ _(, . . . , k) _(d) , b_(k) ₁_(, k) ₂ _(, . . . , k) _(d) , and c_(k) ₁ _(, k) ₂ _(, . . . , k) _(d)represent deterministic real or complex values that represents aresponse to the ARV indexed according to the indexing variables k₁, k₂,. . . , k_(d).

represents the set of real numbers.

represents the set of complex numbers.

represents the set of natural numbers. In the RV addition operation “+:”is meant to introduce the expressions.

The RV addition operation presented above is written when Y_(A) andY_(B) are HRVS. In embodiments in which Y_(A) and Y_(B) are LRVS, d_(A)and/or d_(B) may be equal to one. In embodiments in which Y_(A) andY_(B) are QRVS, d_(A) and/or d_(B) may be equal to two. The order ofY_(C) may be the same as a higher order of Y_(A) or Y_(B). Additionally,in the RV addition operation, non-existing coefficients may be treatedas being equal to zero.

The mathematical operations that may be included in the computerinstructions 105 and are supported and defined by the computer programmodule 108 may include an RV subtraction operation. The mathematicaloperation includes the RV subtraction operation of a first RV and asecond RV. Execution of the computer instruction produces a third RVthat may have a particular indefinite value at a particular definiteprobability and the particular RV represents a result distribution andtracks responses to the one or more ARVS.

FIG. 48 presents an example RV subtraction operation. In FIG. 48, theparameters, functions, etc. are as defined above in the RV additionoperation. The equations of FIG. 48 implement standard mathematicalnomenclature and functionality, which may be understood by one withskill in the art with the benefit of the present disclosure. The RVsubtraction operation presented in FIG. 48 is written when Y_(A) andY_(B) are HRVS. In embodiments in which Y_(A) and Y_(B) are LRVS, d_(A)and/or d_(B) may be equal to one. In embodiments in which Y_(A) andY_(B) are QRVS, d_(A) and/or d_(B) may be equal to two. The order ofY_(C) may be the same as a higher order of Y_(A) or Y_(B). Additionally,in the RV subtraction operation, non-existing coefficients may betreated as being equal to zero.

The mathematical operations that may be included in the computerinstructions 105 and are supported and defined by the computer programmodule 108 may include an RV scaling operation. The mathematicaloperation includes the RV scaling operation of an RV and a constant.Execution of the computer instruction produces another RV that may havea particular indefinite value at a particular definite probability andthe RV represents a result distribution and tracks response to the oneor more ARVS.

FIG. 49 presents an example RV scaling operation. In FIG. 49, theparameters, functions, etc. are as defined above in the RV additionoperation. The parameter b₀ is the constant. The equations of FIG. 49implement standard mathematical nomenclature and functionality, whichmay be understood by one with skill in the art with the benefit of thepresent disclosure. The RV scaling operation presented in FIG. 49 iswritten when Y_(A) is an HRV. In embodiments in which Y_(A) is an LRVS,d_(A) may be equal to one. In embodiments in which Y_(A) is a QRVS,d_(A) may be equal to two. The order of Y_(C) may be the same Y_(A).Division by a constant may be calculated as multiplication with theinverse of the constant.

The mathematical operations that may be included in the computerinstructions 105 and are supported and defined by the computer programmodule 108 may include an RV multiplication operation. For example, themathematical operation includes an RV multiplication operation thatinvolves two of the RVS. The RV addition operation may be definedaccording to example RV multiplication operation expressions:×:Y _(A) +Y _(B) →Y _(C); such that:

$\mspace{76mu}{{Y_{A} = {\sum\limits_{k_{1} = 0}^{m}\;{\ldots{\sum\limits_{k_{\alpha} = 0}^{m}\;{a_{k_{1},{\cdots\; k_{\alpha}}}X_{k_{1}}\ldots\mspace{11mu} X_{k_{\alpha}}}}}}};}$$\mspace{76mu}{{Y_{B} = {\sum\limits_{l_{1} = 0}^{m}\;{\ldots{\sum\limits_{l_{\beta} = 0}^{m}\;{b_{l_{1},{\cdots\; l_{\beta}}}X_{l_{1}}\ldots\mspace{11mu} X_{l_{\beta}}}}}}};}$${Y_{C} \equiv {\sum\limits_{k_{1} = 0}^{m}\;{\ldots{\sum\limits_{k_{\alpha} = 0}^{m}\;{\sum\limits_{l_{1} = 0}^{m}\;{\ldots{\sum\limits_{l_{\beta} = 0}^{m}{c_{k_{1},{\cdots\; k_{\alpha}},l_{1},\cdots,l_{\alpha}}X_{k_{1}}\ldots\mspace{11mu} X_{k_{\alpha}}X_{l_{1}}\mspace{11mu}\ldots\mspace{11mu} X_{l_{\beta}}}}}}}}}};$     c_(k₁, ⋯, k_(α), l₁, ⋯l_(β)) ≡ a_(k₁, ⋯ k_(α))b_(l₁, ⋯ l_(β)); and     m ∈ ℕ.In the RV multiplication operation expressions, Y_(A), Y_(B), and Y_(C)represent RVS. The Y_(A), Y_(B), and Y_(C) are HRVS. The HRV representedby Y_(A) is a αth-order HRV. The HRV represented by Y_(B) is a βth-orderHRV. The HRV represented by Y_(C) is an (α+β)th-order HRV. Theparameters k₁ . . . k_(α) and l₁ . . . l_(β) represent indexingvariables. The parameters X_(k) _(i) and X_(l) _(ij) represents ARVSindexed according to the indexing variables k_(i) and l_(j). Theparameters a_(k) ₁ _(, . . . , k) _(α) , b_(l) ₁ _(, . . . , l) _(β) ,and c_(k) ₁ _(, . . . , k) _(α) _(, l) ₁ _(, . . . , l) _(β) representdeterministic real or complex values that represents a response to theARVS indexed according to the indexing variables k₁, . . . , k_(α) andl₁, . . . , l_(β) and k₁, . . . , k_(α), l₁, . . . , l_(β).

represents the set of real numbers.

represents the set of complex numbers.

represents the set of natural numbers.

One or more of the HRV in the RV multiplication operation expressionsmay be reduced to the canonical form according to the equationspresented in FIGS. 36A-36C. After the reduction to the canonical form,the order of the HRV represented by Y_(C) may be less than m. Inaddition, one or more of the HRV in the RV multiplication operationexpressions may be reduced to lower order RVS using equations related tothe local value assignment presented in FIGS. 43A-43B.

In addition to the RV multiplication operation that involves the HRVS,the mathematical operations that may be included in the computerinstructions 105 and are supported and defined by the computer programmodule 108 may include an RV multiplication operation that involve LRVSand QRVS. FIG. 50 presents RV multiplication operations that involveLRVS. FIG. 51 presents RV multiplication operations that involve QRVS.

In FIGS. 50-51, the parameters, functions, etc. are as defined above inthe RV multiplication operations for the HRVS. The equations of FIGS.50-51 implement standard mathematical nomenclature and functionality,which may be understood by one with skill in the art with the benefit ofthe present disclosure. In the RV multiplication operations for the LRVSof FIG. 50, a product of two LRVS becomes a QRV. In the RVmultiplication operations for the QRVS of FIG. 51, a product of two QRVSbecomes a fourth order HRV.

Conditional Random Variables with Respect to ARV

CRVS may be defined and supported by the computer program module 108.The CRVS may be defined with respect to ARVS. The computer programmodule 108 may define CRVS for the LRVS, CRVS for the QRVS, and CRVS forthe HRVS. FIG. 52 presents a definition for the CRVS for the LRVS. FIG.53 presents a definition for the CRVS for the QRVS. FIG. 54 presents adefinition for the CRVS for the HRVS. The parameters, functions, etc. ofFIG. 52 are as defined above in the LRV expressions. The parameters,functions, etc. of FIG. 53 are as defined above in the QRV expressions.The parameters, functions, etc. of FIG. 53 are as defined above in theHRV expressions. The equations of FIGS. 52-53 implement standardmathematical nomenclature and functionality, which may be understood byone with skill in the art with the benefit of the present disclosure.

Derivative Operator

The mathematical operations that may be included in the computerinstructions 105 and are supported and defined by the computer programmodule 108 may include an RV derivative operation. The RV derivativeoperation is defined according to RV derivative operation expressions:

$\left. {\frac{d}{{dS}_{x}}\text{:}\frac{d}{{dS}_{x}}Y_{A}}\rightarrow Y_{B} \right.;$such that:Y _(B)≡(Y _(A) |X _(i)=1 for ∀i∈S _(X))−(Y _(A) |X _(i)=0 for ∀i∈S_(X)).In the RV derivative operation expressions, Y_(A) and Y_(B) representsRVS. The operator ∀ represents a “for all” operator. The operator |represents a “such that” operator.

The computer program module 108 may define RV derivative operations forthe LRVS, the QRVS, and the HRVS. FIGS. 55A and 55B present a definitionfor RV derivative operations for the LRVS. FIGS. 56A and 56B present adefinition for RV derivative operations for the QRVS. FIGS. 57A and 57Bpresent a definition for RV derivative operations for the HRVS. Theparameters, functions, etc. of FIGS. 55A and 55B are as defined above inthe LRV expressions and the RV derivative operation expressions. Theparameters, functions, etc. of FIGS. 56A and 56B are as defined above inthe QRV expressions and the RV derivative operation expressions. Theparameters, functions, etc. of FIGS. 57A and 57B are as defined above inthe HRV expressions and the RV derivative operation expressions. Theequations of FIGS. 55A-57B implement standard mathematical nomenclatureand functionality, which may be understood by one with skill in the artwith the benefit of the present disclosure.

The computer program module 108 may define multiple statisticalproperties for the RV derivatives. The statistical properties for the RVderivatives may include a ground, an LRV mean, a QRV mean, an HRV mean,an LRV variance, a QRV variance, an HRV variance, an LRV covariance, anda QRV covariance. FIGS. 58A-66C provide equations that may apply to theRV derivative for one or more of the statistical properties. In FIGS.58A-66C, the parameters, functions, etc. are as defined above in the LRVexpressions, the HRV expressions, the QRV expressions, and the RVderivative operation expressions. The equations of FIGS. 58A-66Cimplement standard mathematical nomenclature and functionality, whichmay be understood by one with skill in the art with the benefit of thepresent disclosure.

FIG. 58A presents an example expression that may define an LRV ground ofthe RV derivatives. FIG. 58B presents an example expression that maydefine a QRV ground of the RV derivatives. FIG. 58C presents an exampleexpression that may define an HRV ground of the RV derivatives. FIG. 59presents an example expression that may define an LRV mean of the RVderivatives. FIG. 60 presents example expressions that define a QRV meanof the RV derivatives. FIG. 61 presents example expressions that definean HRV mean of the RV derivatives. FIG. 62 presents an exampleexpression that may define an LRV variance of the RV derivatives. FIGS.63A and 63B present example expressions that define a QRV variance ofthe RV derivatives. FIGS. 64A-64D present example expressions thatdefine an HRV variance of the RV derivatives. FIG. 65 presents anexample expression that may define an LRV covariance of the RVderivatives. FIG. 66C presents example expressions that may define a QRVcovariance of the RV derivatives.

Entropy may be approximated for each of the ARV, the LRV, the QRV, andthe HRV. For instance, FIG. 67 presents an example expression that maybe used to approximate entropy of the ARV. FIGS. 68A-68C present anexample expression for entropy of the LRV. FIGS. 69A-69C present anexample expression for entropy of the QRV; and FIGS. 70A-70C present anexample expression for entropy of the HRV. In FIGS. 68A-70C, Ŝ[Y_(A)]represents an alternative expression for entropy of the LRV, the QRV,and the HRV, respectively.

In FIGS. 68A-70C represent only example entropy expressions for the ARV,the LRV, the QRV, and the HRV. In some embodiments, some otherexpressions may be used to define the entropy. For instance, in someembodiments, one or more weight coefficients (e.g., a_(k) ² and ξ_(k) ²)of entropy may be inverted. Additionally or alternatively, in someexpressions some or all of the weight coefficients may be equal to one.Use of the entropy may be advantageous for global optimization, becauseit is a convex function.

Modifications, additions, or omissions may be made to the programminglanguage system 100 without departing from the scope of the presentdisclosure. For example, the present disclosure applies to a programminglanguage system 100 that may include one or more client devices 102, oneor more server devices 106, one or more networks 104, just the clientdevice 102 and/or the server device 106 (e.g., without the network 104and the server device 106), or any combination thereof.

Moreover, the separation of various components in the embodimentsdescribed herein is not meant to indicate that the separation occurs inall embodiments. It may be understood with the benefit of thisdisclosure that the described components may be integrated together in asingle component or separated into multiple components. For example, thecomputer program module 108 and the application module 107 may beintegrated into a single module.

FIG. 2 illustrates an example computing system 200 configured forquantum-inspired computing in a programming language system. Thecomputing system 200 may be implemented in the programming languagesystem 100 of FIG. 1, for instance. Examples of the computing system 200may include the server device 106 and/or the client device 102. Thecomputing system 200 may include one or more processors 204, a memory206, a communication unit 208, a user interface device 210, and a datastorage 202 that includes the computer program module 108 and/or theapplication module 107 (collectively, modules 108/107).

The processor 204 may include any suitable special-purpose orgeneral-purpose computer, computing entity, or processing deviceincluding various computer hardware or software modules and may beconfigured to execute instructions stored on any applicablecomputer-readable storage media. For example, the processor 204 mayinclude a microprocessor, a microcontroller, a digital signal processor(DSP), an ASIC, an FPGA, or any other digital or analog circuitryconfigured to interpret and/or to execute program instructions and/or toprocess data.

Although illustrated as a single processor in FIG. 2, the processor 204may more generally include any number of processors configured toperform individually or collectively any number of operations describedin the present disclosure. Additionally, one or more of the processors204 may be present on one or more different electronic devices orcomputing systems. In some embodiments, the processor 204 may interpretand/or execute program instructions and/or process data stored in thememory 206, the data storage 202, or the memory 206 and the data storage202. In some embodiments, the processor 204 may fetch programinstructions from the data storage 202 and load the program instructionsin the memory 206. After the program instructions are loaded into thememory 206, the processor 204 may execute the program instructions.

The memory 206 and the data storage 202 may include computer-readablestorage media for carrying or having computer-executable instructions ordata structures stored thereon. Such computer-readable storage media mayinclude any available media that may be accessed by a general-purpose orspecial-purpose computer, such as the processor 204. By way of example,and not limitation, such computer-readable storage media may includetangible or non-transitory computer-readable storage media includingRAM, ROM, EEPROM, CD-ROM or other optical disk storage, magnetic diskstorage or other magnetic storage devices, flash memory devices (e.g.,solid state memory devices), or any other storage medium which may beused to carry or store desired program code in the form ofcomputer-executable instructions or data structures and that may beaccessed by a general-purpose or special-purpose computer. Combinationsof the above may also be included within the scope of computer-readablestorage media. Computer-executable instructions may include, forexample, instructions and data configured to cause the processor 204 toperform a certain operation or group of operations.

The communication unit 208 may include one or more pieces of hardwareconfigured to receive and send communications. In some embodiments, thecommunication unit 208 may include one or more of an antenna, a wiredport, and modulation/demodulation hardware, among other communicationhardware devices. In particular, the communication unit 208 may beconfigured to receive a communication from outside the computing system200 and to present the communication to the processor 204 or to send acommunication from the processor 204 to another device or network (e.g.,104 of FIG. 1).

The user interface device 210 may include one or more pieces of hardwareconfigured to receive input from and/or provide output to a user. Insome embodiments, the user interface device 210 may include one or moreof a speaker, a microphone, a display, a keyboard, a touch screen, or aholographic projection, among other hardware devices.

The modules 108/107 may include program instructions stored in the datastorage 202. The processor 204 may be configured to load the modules108/107 into the memory 206 and execute the modules 108/107.Alternatively, the processor 204 may execute the modules 108/107line-by-line from the data storage 202 without loading them into thememory 206. When executing the modules 108/107, the processor 204 may beconfigured to perform a participation verification process as describedelsewhere in this disclosure.

Modifications, additions, or omissions may be made to the computingsystem 200 without departing from the scope of the present disclosure.For example, in some embodiments, the computing system 200 may notinclude the user interface device 210. In some embodiments, thedifferent components of the computing system 200 may be physicallyseparate and may be communicatively coupled via any suitable mechanism.For example, the data storage 202 may be part of a storage device thatis separate from a server, which includes the processor 204, the memory206, and the communication unit 208, that is communicatively coupled tothe storage device.

FIG. 3 is a flow diagram of an example method 300 of quantum-inspiredcomputing, arranged in accordance with at least one embodiment describedherein. The method 300 may be performed in a programming language systemsuch as the programming language system 100 of FIG. 1. The method 300may be performed in some embodiments by the client device 102, theserver device 106, the computer program module 108, the applicationmodule 107, or another computer system 200 described with reference toFIGS. 1 and 2.

In some embodiments, the client device 102 and/or the server device 106or another computing system may include or may be communicativelycoupled to a non-transitory computer-readable medium (e.g., the memory206 of FIG. 2) having stored thereon programming code or instructionsthat are executable by one or more processors (such as the processor 204of FIG. 2) to cause a computing system and/or the client device 102and/or the server device 106 to perform or control performance of themethod 300. Additionally or alternatively, the client device 102 and/orthe server device 106 may include the processor 204 described elsewherein this disclosure that is configured to execute computer instructionsto cause the client device 102 and/or the server device 106 or anothercomputing system to perform or control performance of the method 300.Although illustrated as discrete blocks, various blocks in FIG. 3 may bedivided into additional blocks, combined into fewer blocks, oreliminated, depending on the desired implementation.

The method 300 may begin at block 302 in which a first ARV may bedefined in a programming language system. The first ARV may have anon-deterministic value of either zero or one. The first ARV may have afirst probability of having a value of one. The first ARV may have asecond probability of having a value of zero. A sum of the firstprobability and the second probability may be one. A covariance of thefirst ARV and a second ARV may be zero.

In some embodiments, a mean the first ARV may be equal to the firstprobability. A mean square of the first ARV may be equal to the firstprobability. An n-th moment of the first ARV may be equal to the firstprobability for all values of n. A variance of the first ARV is equal toa product of the first probability and the second probability. An m-thpower of the first ARV equal to the first ARV for all values of m. Inthese and other embodiments, the first ARV may be defined according toARV expressions provided elsewhere in this disclosure.

At block 304, a first RV may be defined in the programming languagesystem. The first RV may have a first indefinite value at a firstdefinite probability and may include a polynomial of one or more atomicrandom variables (ARVS) that includes the first ARV.

In some embodiments, the first random variable is an LRV. The LRV mayinclude a linear sum of a particular number of the one or more ARVS anda distribution of up to two raised to the power of the particular numberpossible values that correspond to zero or 1 of each of the ARVS. Inthese and other embodiments, the LRV may be defined according to the LRVexpressions provided elsewhere in this disclosure. In some embodiments,the first random variable may be a QRV. In these and other embodiments,the QRV may be defined according to the QRV expressions providedelsewhere in this disclosure. In some embodiments, the first randomvariable may be an HRV. In these and other embodiments, the HRV may bedefined according to the HRV expressions provided elsewhere in thisdisclosure.

At block 306, a computer instruction may be executed that includes amathematical operation that involves the RV as a basic data type.Execution of the computer instruction may produce a second RV that has asecond indefinite value at a second definite probability and the secondRV represents a result distribution and tracks responses to the one ormore ARVS.

In some embodiments, the mathematical operation may include an RVaddition operation of the first RV with a third RV. In these and otherembodiments, the RV addition operation may be defined according to RVaddition operation expressions provided elsewhere in this disclosure. Insome embodiments, the mathematical operation may include an RVmultiplication operation of the first RV with a third RV. In these andother embodiments, the RV multiplication operation may be definedaccording to RV multiplication operation expressions provided elsewherein this disclosure. In some embodiments, the mathematical operationincludes an RV derivative operation of the first RV. In these and otherembodiments, the RV derivative operation may be defined according to RVderivative operation expressions provided elsewhere in this disclosure.

In some embodiments, the computer instruction may be included in anapplication implemented as a quantum simulation for a drug designapplication, a quantum simulation for a material design application, acombinatory optimization for a place and a route in computer aideddesign (CAD), a statistical eye simulation of signal integrity analysisfor high-speed signal transmission, or a statistical static timinganalysis for CAD which explorer the solution space.

One skilled in the art will appreciate that, for this and otherprocedures and methods disclosed herein, the functions performed in theprocesses and methods may be implemented in differing order.Furthermore, the outlined steps and operations are only provided asexamples, and some of the steps and operations may be optional, combinedinto fewer steps and operations, or expanded into additional steps andoperations without detracting from the disclosed embodiments.

The embodiments described herein may include the use of aspecial-purpose or general-purpose computer including various computerhardware or software modules, as discussed in greater detail below.

Embodiments described herein may be implemented using computer-readablemedia for carrying or having computer-executable instructions or datastructures stored thereon. Such computer-readable media may be anyavailable media that may be accessed by a general-purpose orspecial-purpose computer. By way of example, and not limitation, suchcomputer-readable media may include tangible or non-transitorycomputer-readable storage media including RAM, ROM, EEPROM, CD-ROM orother optical disk storage, magnetic disk storage or other magneticstorage devices, or any other non-transitory storage medium which may beused to carry or store desired program code in the form ofcomputer-executable instructions or data structures and which may beaccessed by a general-purpose or special-purpose computer. Combinationsof the above may also be included within the scope of computer-readablemedia.

Computer-executable instructions comprise, for example, instructions anddata, which cause a general-purpose computer, special-purpose computer,or special-purpose processing device to perform a certain function orgroup of functions. Although the subject matter has been described inlanguage specific to structural features and/or methodological acts, itis to be understood that the subject matter defined in the appendedclaims is not necessarily limited to the specific features or actsdescribed above. Rather, the specific features and acts described aboveare disclosed as example forms of implementing the claims.

As used herein, the term “module” or “component” may refer to softwareobjects or routines that execute on the computing system. The differentcomponents, modules, engines, and services described herein may beimplemented as objects or processes that execute on the computing system(e.g., as separate threads). While the system and methods describedherein are preferably implemented in software, implementations inhardware or a combination of software and hardware are also possible andcontemplated. In this description, a “computing entity” may be anycomputing system as previously defined herein, or any module orcombination of modules running on a computing system.

All examples and conditional language recited herein are intended forpedagogical objects to aid the reader in understanding the invention andthe concepts contributed by the inventor to furthering the art, and areto be construed as being without limitation to such specifically recitedexamples and conditions. Although embodiments of the present inventionhave been described in detail, it should be understood that the variouschanges, substitutions, and alterations could be made hereto withoutdeparting from the spirit and scope of the invention.

What is claimed is:
 1. A method of quantum-inspired computing, themethod comprising: defining a first atomic random variable (ARV) in aprogramming language system, the first ARV having a non-deterministicvalue of either zero or one, a first probability of having a value ofone, and a second probability of having a value of zero; a sum of thefirst probability and the second probability is one; and a covariance ofthe first ARV and a second ARV is zero; defining a first random variable(RV) in the programming language system, the first RV having a firstindefinite value at a first definite probability and includes apolynomial of one or more atomic random variables (ARVS) that includesthe first ARV; and executing a computer instruction that includes amathematical operation involving the first RV as a basic data type, theexecuting producing a second RV that has a second indefinite value at asecond definite probability, represents a result distribution, andtracks a response to the one or more ARVS.
 2. The method of claim 1,wherein: a mean of the first ARV is equal to the first probability; amean square of the first ARV is equal to the first probability; an n-thmoment of the first ARV is equal to the first probability for all valuesof n; a variance of the first ARV is equal to a product of the firstprobability and the second probability; and an m-th power of the firstARV is equal to the first ARV for all values of m.
 3. The method ofclaim 1, wherein the first ARV is defined according to ARV expressions:X _(i)∈{0,1};0≤p _(i)≤1;Pr[X _(i)=1]=p _(i);Pr[X _(i)=0]=1−p _(i); andCov(X _(i) ,X _(k))=0 for i≠k; in which: i and k represent indexingvariables; p_(i) represents a first probability indexed according to theindexing variable i; X_(i) represents an ARV indexed according to theindexing variable i; X_(k) represents an ARV indexed according to theindexing variable k; Pr[ ] represents a probability function; and Cov( )represents a covariance function.
 4. The method of claim 3, whereinentropy of the ARV is approximated according to an approximate entropyexpression:S[X _(i)]≃ log 2−2q _(i) ²; in which: S[X_(i)] represents the entropy,q_(i) represents p_(i)−½, and log 2 represents a natural log of 2 baseNapier's constant (e).
 5. The method of claim 3, wherein: the firstrandom variable is a quadratic random variable (QRV); and the QRV isdefined according to the QRV expressions:${Y_{A} \equiv {{\overset{\rightarrow}{X}}^{T}A\overset{\rightarrow}{X}}};$${\overset{\rightarrow}{X} \equiv \left\lbrack {X_{0}\mspace{14mu} X_{1}\mspace{14mu} X_{2}\mspace{11mu}\cdots\mspace{11mu} X_{m}} \right\rbrack^{T}};$${A \equiv \begin{bmatrix}a_{0,0} & a_{0,1} & a_{0,2} & \ldots & a_{0,m} \\a_{1,0} & a_{1,1} & a_{1,2} & \ldots & a_{1,m} \\a_{2,0} & a_{2,1} & a_{2,2} & \ldots & a_{2,m} \\\vdots & \vdots & \vdots & \ddots & \vdots \\a_{m,0} & a_{m,1} & a_{m,2} & \ldots & a_{m,m}\end{bmatrix}};$ a_(k, l) ∈ ℝ  (or  ℂ); m ∈ ℕ; and p₀ = 1;  in which: lrepresents an indexing variable; Y_(A) represents the QRV; X_(k)represents an ARV indexed according to the indexing variable k; a_(k,l)represents a deterministic real or complex value that represents aresponse to the ARVS indexed according to the indexing variables k andl;

represents the set of real numbers;

represents the set of complex numbers;

represents the set of natural numbers; and X₀a_(0,0)X₀ is reserved for aconstant term with p₀=1 such that X₀ is equal to
 1. 6. The method ofclaim 3, wherein: the first random variable is a high-order randomvariable (HRV); and the HRV is defined according to HRV expressions:${Y_{A} \equiv {\sum\limits_{k_{1} = 0}^{m}\;{\sum\limits_{k_{2} = 0}^{m}\;{\ldots{\sum\limits_{k_{d} = 0}^{m}\;{a_{k_{1},k_{2},\cdots,k_{d}}X_{k_{1}}X_{k_{2}}\mspace{11mu}\ldots\mspace{11mu} X_{k_{d}}}}}}}};$a_(k₁, k₂, ⋯ , k_(d)) ∈ ℝ  (or  ℂ); m ∈ ℕ; d ≤ m; and p₀ = 1;  in which:Y_(A) represents the HRV; k and d represent indexing variables; X_(k)_(i) represents an ARV indexed according to the indexing variable k_(i);a_(k) ₁ _(, k) ₂ _(, . . . , k) _(d) represents a deterministic real orcomplex value that represents a response to the ARVS indexed accordingto the indexing variables k₁, k₂, . . . , k_(d);

represents the set of real numbers;

represents the set of complex numbers;

represents the set of natural numbers; and a_(0, 0, . . . , 0)X₀X₀ . . .X₀ is reserved for the constant term with p₀=1 such that X₀ is equalto
 1. 7. The method of claim 1, wherein: the mathematical operationincludes an RV addition operation of the first RV with a third RV; andthe RV addition operation is defined according to RV addition operationexpressions:+:Y _(A) +Y _(B) →Y _(C); such that:${Y_{A} = {a_{0} + {\sum\limits_{k_{1} = 1}^{m}\;{a_{k_{1}}X_{k_{1}}}} + {\sum\limits_{k_{1} = 1}^{m - 1}\;{\sum\limits_{k_{2} = {k_{1} + 1}}^{m}\;{a_{k_{1},k_{2}}X_{k_{1}}X_{k_{2}}}}} + \ldots + {\sum\limits_{k_{1} = 1}^{m - d_{A} + 1}\;{\sum\limits_{k_{2} = {k_{1} + 1}}^{m - d_{A} + 2}\;{\sum\limits_{k_{3} = {k_{2} + 1}}^{m - d_{A} + 3}\;{\cdots{\sum\limits_{k_{d_{A}} = {k_{d_{A - 1}} + 1}}^{m}\;{a_{k_{1},{k_{2}\cdots},k_{d_{A}}}X_{k_{1}}X_{k_{2}}\mspace{11mu}\ldots\mspace{11mu} X_{k_{d_{A}}}}}}}}}}};$${Y_{B} = {b_{0} + {\sum\limits_{k_{1} = 1}^{m}\;{b_{k_{1}}X_{k_{1}}}} + {\sum\limits_{k_{1} = 1}^{m - 1}\;{\sum\limits_{k_{2} = {k_{1} + 1}}^{m}\;{b_{k_{1},k_{2}}X_{k_{1}}X_{k_{2}}}}} + \ldots + {\sum\limits_{k_{1} = 1}^{m - d_{B} + 1}\;{\sum\limits_{k_{2} = {k_{1} + 1}}^{m - d_{B} + 2}\;{\sum\limits_{k_{3} = {k_{2} + 1}}^{m - d_{B} + 3}\;{\ldots{\sum\limits_{k_{d_{B}} = {k_{d_{B - 1}} + 1}}^{m}\;{b_{k_{1},{k_{2}\cdots},k_{d_{B}}}X_{k_{1}}X_{k_{2}}\mspace{11mu}\ldots\mspace{11mu} X_{k_{d_{B}}}}}}}}}}};$${Y_{C} \equiv {c_{0} + {\sum\limits_{k_{1} = 1}^{m}\;{c_{k_{1}}X_{k_{1}}}} + {\sum\limits_{k_{1} = 1}^{m - 1}\;{\sum\limits_{k_{2} = {k_{1} + 1}}^{m}\;{c_{k_{1},k_{2}}X_{k_{1}}X_{k_{2}}}}} + \ldots + {\sum\limits_{k_{1} = 1}^{m - d_{C} + 1}\;{\sum\limits_{k_{2} = {k_{1} + 1}}^{m - d_{C} + 2}\;{\sum\limits_{k_{3} = {k_{2} + 1}}^{m - d_{C} + 3}\;{\ldots{\sum\limits_{k_{d_{C}} = {k_{d_{C - 1}} + 1}}^{m}\;{C_{k_{1},{k_{2}\cdots},k_{d_{C}}}X_{k_{1}}X_{k_{2}}\mspace{11mu}\ldots\mspace{11mu} X_{k_{d_{C}}}}}}}}}}};$     d_(A), d_(B) ≤ m;      d_(C) ≡ max (d_(A), d_(B));     a_(k₁, k₂, ⋯ k_(d_(A))) ∈ ℝ  (or  ℂ);     b_(k₁, k₂, ⋯ k_(d_(B))) ∈ ℝ  (or  ℂ);      m ∈ ℕ;     c_(o) ≡ a₀ + b₀; and$\mspace{76mu}{c_{k_{1},{k_{2}\cdots},k_{l}} \equiv \left\{ {\begin{matrix}{a_{k_{1},{k_{2}\cdots},k_{l}} + b_{k_{1},{k_{2}\cdots},k_{l}}} & {{{for}\mspace{14mu}\left( {1 \leq l \leq d_{A}} \right)} ⩓ \left( {1 \leq l \leq d_{B}} \right)} \\a_{k_{1},{k_{2}\cdots},k_{l}} & {{for}\mspace{14mu}\left( {d_{B} < l \leq d_{A}} \right)} \\b_{k_{1},{k_{2}\cdots},k_{l}} & {{for}\mspace{14mu}\left( {d_{A} < l \leq d_{B}} \right)}\end{matrix};} \right.}$ in which: Y_(A), Y_(B), and Y_(C) representsthe first RV, the third RV, and the second RV, respectively; the Y_(A),Y_(B), and Y_(C) are high-order random variable (HRV); k_(i) representsan indexing variable; X_(k) _(i) represents an ARV indexed according tothe indexing variable k_(i); a_(k) ₁ _(, k) ₂ _(, . . . , k) _(l) ,b_(k) ₁ _(, k) ₂ _(, . . . , k) _(l) , and c_(k) ₁ _(, k) ₂_(, . . . , k) _(l) represent deterministic real or complex values thatrepresents a response to the ARVS indexed according to the indexingvariables k₁ . . . k_(l);

represents the set of real numbers;

represents the set of complex numbers; and

represents the set of natural numbers.
 8. The method of claim 1,wherein: the mathematical operation includes an RV multiplicationoperation of the first RV with a third RV; and the RV multiplicationoperation is defined according to RV multiplication operationexpressions:×:Y _(A) +Y _(B) →Y _(C); such that:$\mspace{76mu}{{Y_{A} = {\sum\limits_{k_{1} = 0}^{m}\;{\ldots{\sum\limits_{k_{\alpha} = 0}^{m}\;{a_{k_{1},{\cdots\; k_{\alpha}}}X_{k_{1}}\ldots\mspace{11mu} X_{k_{\alpha}}}}}}};}$$\mspace{76mu}{{Y_{B} = {\sum\limits_{l_{1} = 0}^{m}\;{\ldots{\sum\limits_{l_{\beta} = 0}^{m}\;{b_{l_{1},{\cdots\; l_{\beta}}}X_{l_{1}}\ldots\mspace{11mu} X_{l_{\beta}}}}}}};}$${Y_{C} \equiv {\sum\limits_{k_{1} = 0}^{m}\;{\ldots{\sum\limits_{k_{\alpha} = 0}^{m}\;{\sum\limits_{l_{1} = 0}^{m}\;{\ldots{\sum\limits_{l_{\beta} = 0}^{m}{c_{k_{1},{\cdots\; k_{\alpha}},l_{1},\cdots,l_{\alpha}}X_{k_{1}}\ldots\mspace{11mu} X_{k_{\alpha}}X_{l_{1}}\mspace{11mu}\ldots\mspace{11mu} X_{l_{\beta}}}}}}}}}};$     c_(k₁, ⋯, k_(α), l₁, ⋯l_(α)) ≡ a_(k₁, ⋯ k_(α))b_(k₁, ⋯ l_(β)); and     m ∈ ℕ;  in which: Y_(A), Y_(B), and Y_(C) represents the first RV,the third RV, and the second RV, respectively; the Y_(A), Y_(B), andY_(C) are high-order random variables (HRV); the Y_(A) is an αth-orderHRV; the Y_(B) is an βth-order HRV; the Y_(C) is an (α-β)th-order HRV;k₁ . . . k_(α) and l₁ . . . l_(β) represent indexing variables; X_(k)_(i) represents an ARV indexed according to the indexing variablesk_(i); X_(l) _(i) represents an ARV indexed according to the indexingvariables l_(i); a_(k) ₁ _(, . . . , k) _(α) , b_(l) ₁ _(, . . . , l)_(β) , and c_(k) ₁ _(, . . . , k) _(α) _(, l) ₁ _(, . . . , l) _(β)represent deterministic real or complex values that represent a responseto the ARV indexed according to the indexing variables k₁ . . . k_(α)and l₁ . . . l_(β);

represents the set of real numbers;

represents the set of complex numbers; and

represents the set of natural numbers.
 9. The method of claim 1,wherein: the mathematical operation includes an RV derivative operationof the first RV; and the RV derivative operation is defined according toRV derivative operation expressions:$\left. {\frac{d}{{dS}_{x}}\text{:}\frac{d}{{dS}_{x}}Y_{A}}\rightarrow Y_{B} \right.;$ such that:Y _(B)≡(Y _(A) |X _(i)=1 for ∀i∈S _(X))−(Y _(A) |X _(i)=0 for ∀i∈S_(X)); in which: Y_(A) and Y_(B) represents the first RV and the secondRV, respectively; ∀ represents a “for all” operator; and | represents a“such that” operator.
 10. The method of claim 1, wherein the computerinstruction is included in an application implemented as: a quantumsimulation for a drug design application; a quantum simulation for amaterial design application; a combinatory optimization for a place anda route in computer aided design (CAD); a statistical eye simulation ofsignal integrity analysis for high-speed signal transmission, or astatistical static timing analysis for CAD which explores the solutionspace.
 11. A non-transitory computer-readable medium having encodedtherein programming code executable by one or more processors to performoperations comprising: defining a first atomic random variable (ARV) ina programming language system, the first ARV having a non-deterministicvalue of either zero or one, a first probability of having a value ofone, and a second probability of having a value of zero; a sum of thefirst probability and the second probability is one; and a covariance ofthe first ARV and a second ARV is zero; defining a first random variable(RV) in the programming language system, the first RV having a firstindefinite value at a first definite probability and includes apolynomial of one or more atomic random variables (ARVS) that includesthe first ARV; and executing a computer instruction that includes amathematical operation involving the first RV as a basic data type, theexecuting producing a second RV that has a second indefinite value at asecond definite probability, represents a result distribution, andtracks a response to the one or more ARVS.
 12. The non-transitorycomputer-readable medium of claim 11, wherein: a mean of the first ARVis equal to the first probability; a mean square of the first ARV isequal to the first probability; an n-th moment of the first ARV is equalto the first probability for all values of n; a variance of the firstARV is equal to a product of the first probability and the secondprobability; and an m-th power of the first ARV is equal to the firstARV for all values of m.
 13. The non-transitory computer-readable mediumof claim 11, wherein the first ARV is defined according to ARVexpressions:X _(i)∈{0,1};0≤p _(i)≤1;Pr[X _(i)=1]=p _(i);Pr[X _(i)=0]=1−p _(i); andCov(X _(i) ,X _(k))=0 for i≠k; in which: i and k represent indexingvariables; p_(i) represents a first probability indexed according to theindexing variable i; X_(i) represents an ARV indexed according to theindexing variable i; X_(k) represents an ARV indexed according to theindexing variable k; Pr[ ] represents a probability function; and Cov( )represents a covariance function.
 14. The non-transitorycomputer-readable medium of claim 13, wherein entropy of the ARV isapproximated according to an approximate entropy expression:S[X _(i)]≃ log 2−2q _(i) ²; in which: S[X_(i)] represents the entropy,q_(i) represents p_(i)−½, and log 2 represents a natural log of 2 baseNapier's constant (e).
 15. The non-transitory computer-readable mediumof claim 13, wherein: the first random variable is a quadratic randomvariable (QRV); and the QRV is defined according to the QRV expressions:${Y_{A} \equiv {{\overset{\rightarrow}{X}}^{T}A\overset{\rightarrow}{X}}};$${\overset{\rightarrow}{X} \equiv \left\lbrack {X_{0}\mspace{14mu} X_{1}\mspace{14mu} X_{2}\mspace{11mu}\cdots\mspace{11mu} X_{m}} \right\rbrack^{T}};$${A \equiv \begin{bmatrix}a_{0,0} & a_{0,1} & a_{0,2} & \ldots & a_{0,m} \\a_{1,0} & a_{1,1} & a_{1,2} & \ldots & a_{1,m} \\a_{2,0} & a_{2,1} & a_{2,2} & \ldots & a_{2,m} \\\vdots & \vdots & \vdots & \ddots & \vdots \\a_{m,0} & a_{m,1} & a_{m,2} & \ldots & a_{m,m}\end{bmatrix}};$ a_(k, l) ∈ ℝ  (or  ℂ); m ∈ ℕ; and p₀ = 1;  in which: lrepresents an indexing variable; Y_(A) represents the QRV; X_(k)represents an ARV indexed according to the indexing variable k; a_(k,l)represents a deterministic real or complex value that represents aresponse to the ARVS indexed according to the indexing variables k;

represents the set of real numbers;

represents the set of complex numbers;

represents the set of natural numbers; and X₀a_(0,0)X₀ is reserved for aconstant term with p₀=1 such that X₀ is equal to
 1. 16. Thenon-transitory computer-readable medium of claim 13, wherein: the firstrandom variable is a high-order random variable (HRV); and the HRV isdefined according to HRV expressions:${Y_{A} \equiv {\sum\limits_{k_{1} = 0}^{m}\;{\sum\limits_{k_{2} = 0}^{m}\;{\ldots{\sum\limits_{k_{d} = 0}^{m}\;{a_{k_{1},k_{2},\cdots,k_{d}}X_{k_{1}}X_{k_{2}}\mspace{11mu}\ldots\mspace{11mu} X_{k_{d}}}}}}}};$a_(k₁, k₂, ⋯ k_(d)) ∈ ℝ  (or  ℂ); m ∈ ℕ; d ≤ m; and p₀ = 1;  in which:Y_(A) represents the HRV; k and d represent indexing variables; X_(k)_(i) represents an ARV indexed according to the indexing variable k_(i);a_(k) ₁ _(, k) ₂ _(, . . . , k) _(d) represents a deterministic real orcomplex value that represents a response to the ARVS indexed accordingto the indexing variables k₁, k₂, . . . , k_(d);

represents the set of real numbers;

represents the set of complex numbers;

represents the set of natural numbers; and a_(0, 0, . . . , 0)X₀X₀ . . .X₀ is reserved for the constant term with p₀=1 such that X₀ is equalto
 1. 17. The non-transitory computer-readable medium of claim 11,wherein: the mathematical operation includes an RV addition operation ofthe first RV with a third RV; and the RV addition operation is definedaccording to RV addition operation expressions:+:Y _(A) +Y _(B) →Y _(C); such that:${Y_{A} = {a_{0} + {\sum\limits_{k_{1} = 1}^{m}\;{a_{k_{1}}X_{k_{1}}}} + {\sum\limits_{k_{1} = 1}^{m - 1}\;{\sum\limits_{k_{2} = {k_{1} + 1}}^{m}\;{a_{k_{1},k_{2}}X_{k_{1}}X_{k_{2}}}}} + \ldots + {\sum\limits_{k_{1} = 1}^{m - d_{A} + 1}\;{\sum\limits_{k_{2} = {k_{1} + 1}}^{m - d_{A} + 2}\;{\sum\limits_{k_{3} = {k_{2} + 1}}^{m - d_{A} + 3}\;{\ldots{\sum\limits_{k_{d_{A}} = {k_{d_{A - 1}} + 1}}^{m}\;{a_{k_{1},{k_{2}\cdots},k_{d_{A}}}X_{k_{1}}X_{k_{2}}\mspace{11mu}\ldots\mspace{11mu} X_{k_{d_{A}}}}}}}}}}};$${Y_{B} = {b_{0} + {\sum\limits_{k_{1} = 1}^{m}\;{b_{k_{1}}X_{k_{1}}}} + {\sum\limits_{k_{1} = 1}^{m - 1}\;{\sum\limits_{k_{2} = {k_{1} + 1}}^{m}\;{b_{k_{1},k_{2}}X_{k_{1}}X_{k_{2}}}}} + \ldots + {\sum\limits_{k_{1} = 1}^{m - d_{B} + 1}\;{\sum\limits_{k_{2} = {k_{1} + 1}}^{m - d_{B} + 2}\;{\sum\limits_{k_{3} = {k_{2} + 1}}^{m - d_{B} + 3}\;{\ldots{\sum\limits_{k_{d_{B}} = {k_{d_{B - 1}} + 1}}^{m}\;{b_{k_{1},{k_{2}\cdots},k_{d_{B}}}X_{k_{1}}X_{k_{2}}\;\ldots\mspace{11mu} X_{k_{d_{B}}}}}}}}}}};$${Y_{C} \equiv {c_{0} + {\sum\limits_{k_{1} = 1}^{m}\;{c_{k_{1}}X_{k_{1}}}} + {\sum\limits_{k_{1} = 1}^{m - 1}\;{\sum\limits_{k_{2} = {k_{1} + 1}}^{m}\;{c_{k_{1},k_{2}}X_{k_{1}}X_{k_{2}}}}} + \ldots + {\sum\limits_{k_{1} = 1}^{m - d_{C} + 1}\;{\sum\limits_{k_{2} = {k_{1} + 1}}^{m - d_{C} + 2}\;{\sum\limits_{k_{3} = {k_{2} + 1}}^{m - d_{C} + 3}\;{\ldots{\sum\limits_{k_{d_{C}} = {k_{d_{C - 1}} + 1}}^{m}\;{C_{k_{1},{k_{2}\cdots},k_{d_{C}}}X_{k_{1}}X_{k_{2}}\;\ldots\mspace{11mu} X_{k_{d_{C}}}}}}}}}}};$     d_(A), d_(B) ≤ m;      d_(C) ≡ max (d_(A), d_(B));     a_(k₁, k₂, ⋯ k_(d_(A))) ∈ ℝ  (or  ℂ);     b_(k₁, k₂, ⋯ k_(d_(B))) ∈ ℝ  (or  ℂ);      m ∈ ℕ;     c_(o) ≡ a₀ + b₀; and$\mspace{76mu}{c_{k_{1},{k_{2}\cdots},k_{l}} \equiv \left\{ {\begin{matrix}{a_{k_{1},{k_{2}\cdots},k_{l}} + b_{k_{1},{k_{2}\cdots},k_{l}}} & {{{for}\mspace{14mu}\left( {1 \leq l \leq d_{A}} \right)} ⩓ \left( {1 \leq l \leq d_{B}} \right)} \\a_{k_{1},{k_{2}\cdots},k_{l}} & {{for}\mspace{14mu}\left( {d_{B} < l \leq d_{A}} \right)} \\b_{k_{1},{k_{2}\cdots},k_{l}} & {{for}\mspace{14mu}\left( {d_{A} < l \leq d_{B}} \right)}\end{matrix};} \right.}$ in which: Y_(A), Y_(B), and Y_(C) representsthe first RV, the third RV, and the second RV, respectively; the Y_(A),Y_(B), and Y_(C) are high-order random variable (HRV); k_(i) representsan indexing variable; X_(k) _(i) represents an ARV indexed according tothe indexing variable k_(i); a_(k) ₁ _(, k) ₂ _(, . . . , k) _(l) ,b_(k) ₁ _(, k) ₂ _(, . . . , k) _(l) , and c_(k) ₁ _(, k) ₂_(, . . . , k) _(l) represent deterministic real or complex values thatrepresents a response to the ARVS indexed according to the indexingvariables k₁ . . . k_(l);

represents the set of real numbers;

represents the set of complex numbers; and

represents the set of natural numbers.
 18. The non-transitorycomputer-readable medium of claim 11, wherein: the mathematicaloperation includes an RV multiplication operation of the first RV with athird RV; and the RV addition operation is defined according to RVmultiplication operation expressions:×:Y _(A) +Y _(B) →Y _(C); such that:$\mspace{76mu}{{Y_{A} = {\sum\limits_{k_{1} = 0}^{m}\;{\ldots{\sum\limits_{k_{\alpha} = 0}^{m}\;{a_{k_{1},{\cdots\; k_{\alpha}}}X_{k_{1}}\ldots\mspace{11mu} X_{k_{\alpha}}}}}}};}$$\mspace{76mu}{{Y_{B} = {\sum\limits_{l_{1} = 0}^{m}\;{\ldots{\sum\limits_{l_{\beta} = 0}^{m}\;{b_{l_{1},{\cdots\; l_{\beta}}}X_{l_{1}}\ldots\mspace{11mu} X_{l_{\beta}}}}}}};}$${Y_{C} \equiv {\sum\limits_{k_{1} = 0}^{m}\;{\ldots{\sum\limits_{k_{\alpha} = 0}^{m}\;{\sum\limits_{l_{1} = 0}^{m}\;{\ldots{\sum\limits_{l_{\beta} = 0}^{m}{c_{k_{1},{\cdots\; k_{\alpha}},l_{1},\cdots,l_{\alpha}}X_{k_{1}}\ldots\mspace{11mu} X_{k_{\alpha}}X_{l_{1}}\mspace{11mu}{\ldots X}_{l_{\beta}}}}}}}}}};$     c_(k₁, ⋯, k_(α), l₁, ⋯l_(α)) ≡ a_(k₁, ⋯ k_(α))b_(l₁, ⋯ l_(β)); and     m ∈ ℕ;  in which: Y_(A), Y_(B), and Y_(C) represents the first RV,the third RV, and the second RV, respectively; the Y_(A), Y_(B), andY_(C) are high-order random variables (HRV); the Y_(A) is an αth-orderHRV; the Y_(B) is an βth-order HRV; the Y_(C) is an (α-β)th-order HRV;k₁ . . . k_(α) and l₁ . . . l_(β) represent indexing variables; X_(k)_(i) represents an ARV indexed according to the indexing variable k_(i);X_(l) _(i) represents an ARV indexed according to the indexing variablel_(i); a_(k) ₁ _(, . . . , k) _(α) , b_(l) ₁ _(, . . . , l) _(β) , andc_(k) ₁ _(, . . . , k) _(α) _(, l) ₁ _(, . . . , l) _(β) representdeterministic real or complex values that represent a response to theARV indexed according to the indexing variables k₁ . . . k_(α) and l₁ .. . l_(β);

represents the set of real numbers;

represents the set of complex numbers; and

represents the set of natural numbers.
 19. The non-transitorycomputer-readable medium of claim 11, wherein: the mathematicaloperation includes an RV derivative operation of the first RV; and theRV derivative operation is defined according to RV derivative operationexpressions:$\left. {\frac{d}{{dS}_{x}}\text{:}\frac{d}{{dS}_{x}}Y_{A}}\rightarrow Y_{B} \right.;$ such that:Y _(B)≡(Y _(A) |X _(i)=1 for ∀i∈S _(X))−(Y _(A) |X _(i)=0 for ∀i∈S_(X)); in which: Y_(A) and Y_(B) represents the first RV and the secondRV, respectively; ∀ represents a “for all” operator; and | represents a“such that” operator.
 20. The non-transitory computer-readable medium ofclaim 11, wherein the computer instruction is included in an applicationimplemented as: a quantum simulation for a drug design application; aquantum simulation for a material design application; a combinatoryoptimization for a place and a route in computer aided design (CAD); astatistical eye simulation of signal integrity analysis for high-speedsignal transmission, or a statistical static timing analysis for CADwhich explores the solution space.